Bruce Hardie: BG/NBD and Customer-Base Analysis Models
- Bruce Hardie is a Professor of Marketing at London Business School. His research focuses on customer and marketing analytics, especially probability models for customer-base analysis.
- The BG/NBD model (Fader, Hardie and Lee, Marketing Science, 2005) forecasts repeat purchases, gives results very similar to the Pareto/NBD, and can be estimated in Microsoft Excel.
- The iso-value curve paper (Journal of Marketing Research, 2005) linked recency, frequency and monetary value to a formal Customer Lifetime Value forecast.
- The right model depends on your business setting: contractual or noncontractual, and continuous or discrete purchase occasions.
- An online store is usually a noncontractual business, so customers leave without telling you and their activity has to be inferred from transaction history.
Bruce Hardie is a Professor of Marketing at London Business School who builds probability models for customer-base analysis. With Peter Fader, he created the BG/NBD and BG/BB models and the iso-value curve approach that links RFM to Customer Lifetime Value. These models help a business predict which customers are still active, how often they will buy and what they are worth.
This article covers who he is, why his work matters for Customer Value Optimization, the papers worth reading, and how to choose and apply the right model for your business.
Who Bruce Hardie is
Bruce Hardie studied in New Zealand before completing his MA in Managerial Science and Applied Economics and his PhD in Marketing at the University of Pennsylvania. He now teaches at London Business School.
His research focuses on developing data-based models that are easy to implement and use by marketing analysts and decision-makers. His earlier work centered on new product sales forecasting and marketing mix analysis. Much of his current work focuses on tools for customer analytics. His research has appeared in numerous marketing, statistics and operations research journals.
Most of his customer analytics work is co-authored with Peter Fader of The Wharton School. Their 1996 Journal of Marketing Research paper, Modeling Consumer Choice Among SKUs, won the 1997 Paul E. Green Award. Hardie also keeps his papers, teaching notes and datasets freely available on his website.
Why his work matters for Customer Value Optimization
Most businesses report the past: revenue last month, orders last quarter. Customer Value Optimization needs a view of the future, customer by customer. That is the problem Hardie's research solves.
The hard part is that most customers do not tell you when they leave. A customer who has not ordered in four months may be gone, or may be between purchases. Hardie and Fader's models use each customer's recency and frequency to estimate the probability that the customer is still active, and then forecast their future purchases. Add a model for spend per transaction and you have a forward-looking Customer Lifetime Value estimate.
Their 2009 overview, Probability Models for Customer-Base Analysis (Journal of Interactive Marketing), sets out a taxonomy of business settings and the models suited to each. It is the best single starting point for his work.
Bruce Hardie's key papers
“Counting Your Customers” the Easy Way: An Alternative to the Pareto/NBD Model
By Peter S. Fader, Bruce G. S. Hardie and Ka Lok Lee, Marketing Science, 2005 (PDF).
In this paper, the authors suggest a new way to predict future customer purchasing patterns. Their beta-geometric/NBD (BG/NBD) model is an alternative to the Pareto/NBD “Counting Your Customers” framework proposed by Schmittlein, Morrison and Colombo. The Pareto/NBD is powerful but hard to estimate, which kept it out of most marketing teams' hands.
“We develop a new model, the beta-geometric/NBD (BG/NBD), which represents a slight variation in the behavioral ‘story’ associated with the Pareto/NBD, but it is vastly easier to implement.”
The parameters of the model can be obtained in Microsoft Excel, and the authors find that the two models give very similar results in a wide variety of purchasing environments. Hardie also publishes a step-by-step note on implementing the BG/NBD model in Excel. Open-source libraries such as PyMC-Marketing implement it.
RFM and CLV: Using Iso-Value Curves for Customer Base Analysis
By Peter S. Fader, Bruce G. S. Hardie and Ka Lok Lee, Journal of Marketing Research, 2005 (PDF).
In this paper, the authors present a formal model that links the RFM model and customer lifetime value. It needs only RFM inputs to project the lifetime value of a group of customers, using the Pareto/NBD model for the flow of transactions and a gamma-gamma submodel for spend per transaction. The key to the analysis is the “iso-value” curve:
“Key to this analysis is the notion of ‘iso-value’ curves, which enable us to group together individual customers who have different behavioral histories but similar future valuations. Iso-value curves make it easy to visualize and summarize the main interactions and tradeoffs among the RFM measures and CLV.”
This is the paper that turned RFM scoring from a reporting habit into a forecasting input.
Customer-Base Analysis in a Discrete-Time Noncontractual Setting
By Peter S. Fader, Bruce G. S. Hardie and Jen Shang, Marketing Science, 2010 (PDF).
In this paper, the authors present a beta-geometric/beta-Bernoulli (BG/BB) model that helps predict future customer behavior for businesses that do not use a subscription model and where transactions happen at discrete occasions, such as an annual event or a yearly donation drive. They apply it to donations made to a public radio station.
The authors say that “the model is easy to implement in a standard spreadsheet environment and yields relatively simple closed-form expressions for the expected number of future transactions conditional on past observed behavior (and other quantities of managerial interest).”
Can We Infer “Trial and Repeat” Numbers From Aggregate Sales Data?
By Peter S. Fader and Bruce G. S. Hardie, working paper, 2003 (PDF).
In this paper, Hardie and Fader show that product performance monitoring and forecasting should be performed with caution when you split sales into trial and repeat purchases. The authors explain their method:
“Using panel data for twenty new products, we aggregate the household-level transaction data to arrive at aggregate sales data. We fit a model of new product sales to these data and compare the implied trial/repeat patterns to the actual patterns observed using the raw panel data.”
They conclude that “any inferences derived from the aggregate sales data using such models can be very misleading.” The lesson carries over directly to customer analytics: totals hide who is buying for the first time and who is coming back. You need customer-level data to see it.
Modeling the Evolution of Repeat Buying
By Peter S. Fader and Bruce G. S. Hardie, working paper, 1999 (PDF).
In this paper, the authors suggest a nonstationary model that can capture and forecast repeat buying that changes over time. It is an alternative to the negative binomial distribution (NBD) model:
“We introduce a model — the nonstationary exponential-gamma (NSEG) model — that accomplishes these tasks while retaining the well-known robustness, interpretability, and other desirable properties of the basic NBD framework. […] We demonstrate that NSEG performs very well on both dimensions, especially in contrast to the benchmark NBD model.”
Further reading
Three more pieces complete the picture. How to Project Customer Retention (Journal of Interactive Marketing, 2007) presents the shifted-beta-geometric model for subscription businesses, which can also be built in Excel. Customer-Base Valuation in a Contractual Setting: The Perils of Ignoring Heterogeneity (Marketing Science, 2010) shows why a single aggregate retention rate undervalues a customer base. And The Customer-Base Audit, written with Peter Fader and Michael Ross and published by Wharton School Press in 2022, explains how to review your customers' buying behavior before you build any model.
Which Hardie model fits your business?
Fader and Hardie's taxonomy sorts businesses by these two questions. The table below maps each setting to the model from the papers above and to what it tells you.
| Business setting | Typical example | Model to look at | What it tells you |
|---|---|---|---|
| Noncontractual, buy at any time | Online store, retail | BG/NBD or Pareto/NBD, plus gamma-gamma for spend | Probability each customer is still active, expected purchases, CLV |
| Noncontractual, discrete occasions | Annual event, yearly donation drive | BG/BB | Expected future transactions from each customer's past pattern |
| Contractual, discrete renewals | Monthly or yearly subscription | Shifted-beta-geometric | Projected retention and customer tenure by cohort |
| Any setting, RFM already in use | Stores with RFM segments | Iso-value curves | Which RFM combinations share the same future value |
How to apply Hardie's research to your customer base
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Build a customer-level transaction tableYou need one row per order with a customer ID, a date and a value. Aggregate sales totals are not enough, as Hardie and Fader's trial-and-repeat paper shows.
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Identify your business settingDecide whether you are contractual or noncontractual, and whether customers can buy at any time or only at discrete occasions. Use the table above to choose the model.
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Calculate recency, frequency and monetary valueThese three inputs feed the BG/NBD, the gamma-gamma spend model and the iso-value curves. See our RFM segmentation guide for how to score them.
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Forecast activity and value per customerEstimate the probability that each customer is still active, their expected purchases and their expected spend. Start with Hardie's free Excel notes if you do not have a data team.
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Act on the segmentsGroup customers by expected value. Protect high-value customers whose probability of being active is falling, and stop spending on customers who are very unlikely to return.
Customer Intelligence in Nexus by Omniconvert automates this work for eCommerce: RFM scoring across the whole customer base, cohort analysis, CLV by segment, and the ability to push high-value or at-risk segments to Meta Ads, Google Ads and Klaviyo. It is built on 13 years of customer data across 7,000+ websites and 15+ industries. To learn the strategy these models support, the CVO Academy teaches the Customer Value Optimization framework.
Frequently Asked Questions
Bruce Hardie is a Professor of Marketing at London Business School. He holds a BCom and an MCom from the University of Auckland and an MA and PhD from the University of Pennsylvania. His work focuses on customer and marketing analytics, in particular probability models for customer-base analysis, and much of it is co-authored with Peter Fader of The Wharton School.
The beta-geometric/NBD (BG/NBD) model forecasts how many purchases each customer will make in the future when customers can stop buying at any time without telling you. Fader, Hardie and Lee introduced it in Marketing Science in 2005 as an alternative to the Pareto/NBD model. It tells a slightly different behavioral story, gives very similar results, and its parameters can be estimated in Microsoft Excel.
Both models forecast repeat purchases in a noncontractual setting. The Pareto/NBD, proposed by Schmittlein, Morrison and Colombo, lets a customer drop out at any moment in time. The BG/NBD assumes a customer can only drop out immediately after a purchase. That small change makes the BG/NBD vastly easier to implement, and Fader, Hardie and Lee found that the two models give very similar results in a wide variety of purchasing environments.
The beta-geometric/beta-Bernoulli (BG/BB) model is for businesses where transactions happen at discrete occasions, such as yearly events or one purchase opportunity per period, and there is no contract. It predicts future transactions from each customer's past pattern of activity. Fader, Hardie and Shang published it in Marketing Science in 2010 and applied it to donations made to a public radio station.
Iso-value curves group customers who have different purchase histories but the same expected future value. They come from the paper RFM and CLV: Using Iso-Value Curves for Customer Base Analysis by Fader, Hardie and Lee, published in the Journal of Marketing Research in 2005. The paper links recency, frequency and monetary value to a formal CLV forecast, using the Pareto/NBD model for transactions and a gamma-gamma submodel for spend.
Hardie and Peter Fader of The Wharton School have co-authored most of the papers covered here, including the BG/NBD, BG/BB, iso-value curve and shifted-beta-geometric work. They also wrote The Customer-Base Audit with Michael Ross, published by Wharton School Press in 2022. Fader is best known for the customer centricity argument, and Hardie for the models that make it measurable.
Yes, for a first pass. Ease of implementation is a stated goal of his research. Hardie publishes free notes on his website that show how to implement the BG/NBD and BG/BB models in Excel, and open-source libraries such as PyMC-Marketing include BG/NBD implementations. The main requirement is clean transaction data at customer level: a customer ID, a date and a value for each order.
Treat an online store as a noncontractual business: customers leave silently, so you have to infer who is still active. Build a customer-level transaction table, calculate recency, frequency and monetary value for each customer, estimate who is likely to buy again and how much, then group customers by expected value and spend accordingly. Nexus by Omniconvert automates the RFM scoring, cohort analysis and CLV tracking this work depends on.
Bruce Hardie's contribution is practical. He took the question every retailer asks, which customers are still with us and what they are worth, and built models that answer it from ordinary transaction data. The BG/NBD, the BG/BB and the iso-value curves all start from the same inputs: when each customer bought, how often, and how much. If you have those three things, you already have what you need to stop treating your customer base as one average and start forecasting it customer by customer.
Forecast the value of every customer
Nexus by Omniconvert scores every customer on recency, frequency and monetary value, tracks CLV by cohort, and pushes your high-value and at-risk segments to Meta Ads, Google Ads and Klaviyo. Built on 13 years of customer data across 7,000+ websites and 15+ industries.